Difference and functional equations Definition

Stability of a difference equation

Builds on Equilibrium of a difference equation Norm

Definition

Let $D \subseteq \R^m$ , let $\norm{\cdot}$ be a norm on $\R^m$ , and let $f \colon D \to D$ be a map. Let $x^* \in D$ be an equilibrium of $x_{n+1} = f(x_n)$ , so $f(x^*) = x^*$ . Write $f^n$ for the $n$ -th iterate of $f$ .

The equilibrium $x^*$ is stable if for every $\varepsilon > 0$ there is $\delta > 0$ such that $\norm{x_0 - x^*} < \delta$ and $x_0 \in D$ imply $\norm{f^n(x_0) - x^*} < \varepsilon$ for all $n \ge 0$ .
It is attracting if there is $\eta > 0$ such that $\norm{x_0 - x^*} < \eta$ and $x_0 \in D$ imply $f^n(x_0) \to x^*$ as $n \to \infty$ .
It is asymptotically stable if it is stable and attracting.
It is unstable if it is not stable.

Remarks

Example: for $f(x) = x/2$ on $\R$ , the equilibrium $0$ is asymptotically stable: take $\delta = \varepsilon$ , since $\abs{f^n(x_0)} = \abs{x_0}/2^n \le \abs{x_0}$ , and $f^n(x_0) \to 0$ for every $x_0$ . Non-example: for $f(x) = 2x$ , the equilibrium $0$ is not stable, since $\abs{f^n(x_0)} = 2^n \abs{x_0}$ exceeds any $\varepsilon$ for $x_0 \ne 0$ . For $f(x) = -x$ , the equilibrium $0$ is stable but not attracting, as solutions jump between $x_0$ and $-x_0$ . Since all norms on $\R^m$ are equivalent, the notions do not depend on the choice of norm. For a higher-order equation, the same definitions apply to the equivalent first-order system on $\R^{mk}$ . They are the discrete analogues of the definitions of Lyapunov stability and asymptotic stability for flows.

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Equilibrium of a difference equation concept
Norm concept
Stability of a difference equation definition

Tag	`9BPTKH2HYJ`
URI	`m2://9BPTKH2HYJ`
Kind	definition
Writing	W0 stub
MSC	39A30
Defines	Stability of a difference equation
Uses	Equilibrium of a difference equation, Norm

Stability of a difference equation

Definition

Remarks

Reading path

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